Operators Compound tensor

The fact that the magnetic interaction Hamiltonians are compound tensor operators can be exploited to derive more specific selection rules than the one given above. Furthermore, as we shall see later, the number of matrix elements between multiplet components that actually have to be computed can be considerably reduced by use of the Wigner-Eckart theorem. 

We are mainly interested in compound tensor operators of rank zero (i.e., scalar operators such as the Hamiltonian). To form a scalar from two tensor operators 0 and /l, their ranks k and j have to be equal. Further, the +q component of lk> has to be combined with the -q component of and... 

Table 8 Irreducible Spherical Compound Tensor Operators Resulting from a Product of Two First-Rank Tensor Operators J T and. 9.T...

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

Full tabulations of the vibronic coupling constants and tensor operators involved in the calculation of f —> f electronic transitions in centrosymmetric compounds are available and can be requested from R.A. (Documentation I - Appendices 1 and 2), and therefore will not be repeated here. 

As an example, consider the product of two arbitrary first-rank tensor operators 0 and It is nine-dimensional and can be reduced to a sum of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin-spin coupling Hamiltonians. In terms of spherical and Cartesian components of 0 and J2, the resulting irreducible tensors are given in Tables 8 and 9, respectively.70... 

The compound irreducible tensor operators of the second rank are constructed from the first-rank tensors (spherical vectors) as follows... 

The superscript refers to the rank of the tensor whereas the subscript distinguishes among its components. While represents the spherical transform of the parameter tensor, B 0 C yi represents the compound operator part constituted of the scalar, vector and tensor products of physical vectors. The important relationships are contained in Table 53. 

---